Mathematics Department Colloquium

March 23 4:15-5:15

Location: SAS 4201

Speaker: Andrew Sageman-Furnas (NC State)

Title: Discovering Isometric Tori With The Same Curvatures

Abstract: A longstanding problem in differential geometry, known as the Bonnet problem, asks: is a compact surface in Euclidean three-space uniquely determined by its metric and mean curvature function? The answer is known to be yes for a topological sphere and yes for a generic surface. In this talk, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. These tori are the first examples of compact Bonnet pairs. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. We describe the discovery of these analytic tori using discrete differential geometry. It involves exploring immersions of a 5×7 quad decomposition of a torus and a theory of discrete Bonnet pairs. The smooth/analytic theory is joint work with Alexander Bobenko and Tim Hoffmann, and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.